Stated another way, a gradient is a vector that has coordinate components that consist of the partial derivatives of a function with respect to each of its variables. For example, if , then

. Observe that in this case, the gradient vector is orthogonal to the "level curve" defined by , which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of , and vectors outward from the origin are perpendicular to circles centered at the origin. We'll see later that this is a case of a more general property of the gradient.

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction is maximal in the direction of the gradient, and equal to for a unit vector in the direction of the gradient.

Properties of the Gradient

If is a differentiable function with smooth level sets, then the gradient vector field is perpendicular to the level sets of . For fix a level set , and let be a vector tangent to at . Then we can find a curve on with . Now

since is a level set. Taking derivatives of both sides and applying the chain rule, we get that

Thus, is perpendicular to at , i.e., the gradient of is perpendicular to the level sets of .